Related papers: The complexity of classification problems for mode…
We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has…
Given a first-order theory $T$ formulated in the usual language of first-order arithmetic, we say that $T$ is of *restricted complexity* if there is some natural number $n$ and some set $\mathcal A$ of $\Sigma_n$-sentences such that $T$ can…
We study fundamental clustering problems for incomplete data. Specifically, given a set of incomplete d-dimensional vectors (representing rows of a matrix), the goal is to complete the missing vector entries in a way that admits a…
We introduce the notion of an arithmetic matroid, whose main example is given by a list of elements of a finitely generated abelian group. In particular we study the representability of its dual, providing an extension of the Gale duality…
The emptiness and containment problems for probabilistic automata are natural quantitative generalisations of the classical language emptiness and inclusion problems for Boolean automata. It is well known that both problems are undecidable.…
A novel correction algorithm is proposed for multi-class classification problems with corrupted training data. The algorithm is non-intrusive, in the sense that it post-processes a trained classification model by adding a correction…
Borel-fixed ideals play a key role in the study of Hilbert schemes. Indeed each component and each intersection of components of a Hilbert scheme contains at least one Borel-fixed point, i.e. a point corresponding to a subscheme defined by…
We prove that the classification problem for graphs and several types of algebraic lattices (distributive, congruence and modular) up to isomorphism contains the classification problem for pairs of matrices up to simultaneous similarity.
We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with Z/2 coefficients an analog…
Motivated by the remarkable interplay between (chordal) graphs and matrix algebra, we associate to each graph a so-called completion number that might encode some aspects of that interplay. We show that this number is not trivial, and we…
We consider pairs of finitely presented, residually finite groups $P\hookrightarrow\G$ for which the induced map of profinite completions $\hat P\to \hat\G$ is an isomorphism. We prove that there is no algorithm that, given an arbitrary…
Robin Hirsch posed in 1996 the 'Really Big Complexity Problem': classify the computational complexity of the network satisfaction problem for all finite relation algebras A. We provide a complete classification for the case that A is…
We consider the problem of computing the Euler characteristic of an abstract simplicial complex given by its vertices and facets. We show that this problem is #P-complete and present two new practical algorithms for computing Euler…
In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of…
We consider two matrix completion problems, in which we are given a matrix with missing entries and the task is to complete the matrix in a way that (1) minimizes the rank, or (2) minimizes the number of distinct rows. We study the…
We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which…
We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the…
We distinguish finitarily between algorithmic verifiability, and algorithmic computability, to show that Goedel's 'formally' unprovable, but 'numeral-wise' provable, arithmetical proposition [(Ax)R(x)] can be finitarily evidenced as:…
The conjugacy problem for a finitely generated group $G$ is the two-variable problem of deciding for an arbitrary pair $(u,v)$ of elements of $G$, whether or not $u$ is conjugate to $v$ in $G$. We construct examples of finitely generated,…
There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem…